Extremal pseudocompact abelian groups: a unified treatment (Q2856927)

This paper addresses questions related to certain extremal phenomena that occur on pseudocompact groups. More precisely, call a pseudocompact group \(G = (G,\mathcal{T})\):NEWLINENEWLINE(a) \(r\)-extremal if no topology on \(G\) strictly finer than \(\mathcal{T}\) makes \(G\) a pseudocompact topological group;NEWLINENEWLINE(b) \(s\)-extremal if \(G\) admits no proper dense pseudocompact subgroup.NEWLINENEWLINEIt was first noticed by the first author and \textit{L. C. Robertson} [Proc. Am. Math. Soc. 86, 173--178 (1982; Zbl 0508.22002)] that metrizable (pseudo)compact abelian groups are both \(s\)-extremal and \(r\)-extremal. They conjectured that these implications can be inverted, namelyNEWLINENEWLINE(\(*\)) abelian non-metrizable pseudocompact groups are neither \(s\)-extremal nor \(r\)-extremal.NEWLINENEWLINEFor torsion groups (\(*\)) was proved by the first author and \textit{L. C. Robertson} [Diss. Math. 272, 42 p. (1988; Zbl 0703.22002)]. This result was extended in two ways by \textit{A. Giordano Bruno} et al. [Appl. Gen. Topol. 7, No. 1, 1--39 (2006; Zbl 1127.22003)]. The first extension was towards a larger class of pseudocompact abelian groups, called by these authors singular, namely the pseudocompact abelian groups having a torsion closed \(G_\delta\)-subgroup. (In the compact case the relevance of these groups was established much earlier by \textit{D. Shakhmatov} and \textit{D. Dikranjan} [Proc. Am. Math. Soc. 114, No. 4, 1119--1129 (1992; Zbl 0781.54018)] -- under the assumption of the Lusin Hypothesis \(2^{\omega _1}=2^{\omega }\) the compact abelian groups that admit a proper totally dense pseudocompact subgroup are precisely the singular ones.) The aforementioned paper of A. Giordano Bruno et al. introduced also a weaker notion of extremality. Namely, a pseudocompact abelian group \(G\) is \textit{\(d\)-extremal} if for every dense pseudocompact subgroup \(H\) of \(G\) the quotient \(G/H\) is divisible. Obviously, \(s\)-extremal abelian groups are \(d\)-extremal; these authors proved that also \(r\)-extremal abelian groups are \(d\)-extremal. In this way, their Corollary 5.17 (stating that \(d\)-extremal singular pseudocompact groups are metrizable) entails, in particular, that (\(*\)) holds for singular pseudocompact groups. Various authors proved that conjecture (\(*\)) is true in some particular cases.NEWLINENEWLINEThe present paper presents a polished, complete and self-contained proof of this remarkable theorem. The first shorter proof presented in [Zbl 1127.22003] had two independent essentially disjoint parts: the case of a torsion group and the non-torsion case. This is why the authors find it appropriate to strive for a unified approach treating simultaneously both cases in the present paper. Nevertheless, the authors use again a bifurcation: this time between the case of singular pseudocompact groups (covered by Theorem 4.1) and the case in which the pseudocompact group is not singular (covered by Theorem 4.2). In the last Section 5 the authors provide a review of the literature.NEWLINENEWLINE Reviewer's note: It is unfortunate that the review of the literature in the last section does not mention the paper of \textit{A. Giordano Bruno} [Forum Math. 21, No. 4, 639--659 (2009; Zbl 1173.22002)] dealing with extremality in the class of \(\alpha \)-pseudocompact groups. (According to \textit{J. F. Kennison} [Trans. Am. Math. Soc. 104, 436--442 (1962; Zbl 0111.35004)], for an infinite cardinal \(\alpha \) a topological space \(X\) is \(\alpha \)-pseudocompact if \(f(X)\) is compact in \(\mathbb{R}^{\alpha }\) for every continuous function \(f\: X\rightarrow \mathbb{R}^{\alpha }.\) Clearly, \(\omega \)-pseudocompactness coincides with pseudocompactness; while \(\alpha \)-pseudocompact spaces of weight \(\leq \alpha \) are compact.) This paper introduces the notions of \(s_{\alpha }\)-extremality, \(r_{\alpha }\)-extremality and \(\alpha \)-singularity for \(\alpha \)-pseudocompact groups in obvious analogy with \(s\)-extremality, \(r\)-extremality and \(\alpha \)-singularity in the pseudocompact case. The main theorem, counterpart of (\(*\)), states: an abelian \(\alpha \)-pseudocompact group has weight \(\leq \alpha \) if and only if it is \(s_{\alpha }\)-extremal if and only if it is \(r_{\alpha }\)-extremal. This proof is based again on the bifurcation between the \(\alpha \)-singular and the non-\(\alpha \)-singular case, realizing in this way the ``unified approach'' to extremality even in the case of \(\alpha \)-pseudocompact groups.